Polynomial interpolation

Read the short section on polynomial interpolation in Ch$.1\mathrm{.}10$ of the textbook and answer

the following question.

Question $1$:

$[4$ pts$]$ Find the cubic polynomial which passes through the following four points:

$(1,1),(0,1),(1,3),(4,1).$

$$ Set up your system of linear equations clearly.

$$ Write your system in an appropriate matrix form.

$$ Solve your system using whatever method you like. Note: you may use MATLAB,

octave, or whatever software you wish; just make sure that this is indicated on the

page, and that the explanation of the previous two steps is clear.

$$ Write down your final answer as a cubic polynomial which passes through those points.

Bonus: $[2$ pts$]$ Demonstrate $($using linear systems$)$ why it is not possible to find a quadratic

polynomial which passes through these four points.

Cryptography

Read the section on cryptography $($specifically$,$ Hill $2-$ciphers$)$ in Ch$.10$ of the textbook.

This is Section $10\mathrm{.}13$ in the $12$th edition, $10\mathrm{.}14$ in the $11$th edition, and $10\mathrm{.}15$ in the $10$th

edition. Answer the following question: you should show all of your work, and all of your

steps to reach the answer. There are several examples in the textbook which you can follow

closely$$show all of the same work that they do$.$

Question $2$:

$($a$)[2$ pts$]$ Use a Hill$-2$ cypher with matrix A $=$

$25$

$32$

to encrypt the message $$DO IT$.$

$($b$)[4$ pts$]$ Decrypt the message $$EPAZ$,$ encrypted using A above; you will have to explicitly

compute A$1($and its entries should be numbers between $0$ and $25)$ and use it in the

decryption.